Given:
The given equation is
[tex]y=x^2+16[/tex]Required:
We need to find the inverse for the equation.
Explanation:
[tex]\text{ Let y=f\lparen x\rparen and }x=f^{-1}(y)\text{ and substitute }x=f^{-1}(y)\text{ in the given equation.}[/tex][tex]y=(f^{-1}(y))^2+16[/tex]Substract 16 from both sides of the equation.
[tex]y-16=(f^{-1}(y))^2+16-16[/tex][tex]y-16=(f^{-1}(y))^2[/tex]Take square root on both sides of the equation.
[tex]\pm\sqrt{(y-16)}=f^{-1}(y)[/tex][tex]f^{-1}(y)=\pm\sqrt{(y-16)}[/tex]Replace y=x in the equation.
[tex]f^{-1}(x)=\pm\sqrt{x-16}[/tex]Final answer:
[tex]f^{-1}(x)=\pm\sqrt{x-16}[/tex]Find the x-intercept and the y-intercept without graphing. Write the coordinates of each intercept. When typing the point (x,y) be sure to include parentheses and a comma between your x and y components. Do not put any spaces between your characters. If a value is not an integer type your answer rounded to the nearest hundredth.3x+8y=24the x-intercept is Answerthe y-intercept is Answer
We want to find the x and y-intercepts of
[tex]3x+8y=24[/tex]The x-intercept is where the graph cuts the x-axis, when y = 0. To find this in our equation, we just need to evaluate it at y = 0.
[tex]\begin{gathered} 3x+8\times0=24 \\ 3x=24 \\ x=\frac{24}{3}=8 \end{gathered}[/tex]Then, the x-intercept is (8, 0).
The y-intercept is where the graph cuts the y-axis, when x = 0. To find this in our equation, we just need to evaluate it at x = 0.
[tex]\begin{gathered} 3\times0+8y=24 \\ 8y=24 \\ y=\frac{24}{8}=3 \end{gathered}[/tex]The y-intercept is (0, 3).
if the point (-1,4)and (2,13)are on the graph of the quadratic function [tex]y = 7x {}^{2} + bx + c[/tex]what are the values of b and c
The Solution:
Given:
[tex]y=7x^2+bx+c[/tex]Given that the points: (-1,4) and (2,13) are on the graph of the given equation,
We are required to find the values of a and b.
Substitute (x= -1, y = 4) in the equation, we get:
[tex]\begin{gathered} 4=7(-1)^2+b(-1)+c \\ 4=7-b+c \\ -3=-b+c...eqn(1) \end{gathered}[/tex]Substitute (x= 2, y = 13) in the equation, we get:
[tex]\begin{gathered} 13=7(2)^2+b(2)+c \\ 13=28+2b+c \\ -15=2b+c...eqn(2) \end{gathered}[/tex]Solving eqn(1) and eqn(2) simultaneously by the elimination method:
Subtract eqn(1) from eqn(2):
[tex]\begin{gathered} -15--3=2b--b+c-c \\ -12=3b \end{gathered}[/tex]Divide both sides by 3.
[tex]b=\frac{-12}{3}=-4[/tex]Substitute -6 for b in eqn(1).
[tex]\begin{gathered} -3=-b+c \\ -3=-(-4)+c \\ \\ -3=4+c \\ -3-4=c \\ -7=c \\ c=-7 \end{gathered}[/tex]Therefore, the correct answers are:
b = -4
c = -7
caluculate the length of AC to 1 decimal place in the trapezium below.
Check the picture below.
usign the pythagorean theorem let's find the side CD, then let's get the side AC using the same pythagorean threorem.
[tex]\textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies \sqrt{c^2 - b^2}=a \qquad \begin{cases} c=\stackrel{hypotenuse}{16}\\ a=\stackrel{adjacent}{CD}\\ b=\stackrel{opposite}{7}\\ \end{cases} \\\\\\ \sqrt{16^2 - 7^2}=CD\implies \sqrt{207}=CD \\\\[-0.35em] ~\dotfill[/tex]
[tex]c^2=a^2+b^2\implies c=\sqrt{a^2 + b^2} \qquad \begin{cases} c=\stackrel{hypotenuse}{AC}\\ a=\stackrel{adjacent}{CD}\\ b=\stackrel{opposite}{11}\\ \end{cases} \\\\\\ AC=\sqrt{(\sqrt{207})^2~~ + ~~11^2}\implies AC=\sqrt{207 + 121}\implies \boxed{AC\approx 18.1}[/tex]
Felipe states that he can use the
inequality 1 ≤ x ≤ 4 to describe the domain
{1, 2, 3, 4} for a given function. Explain Felipe's
error.
For the inequality 1 ≤ x ≤ 4 given by Felipe the domain of the function is stated as {1,2,3,4} which shows the error of x belongs to which set of numbers is not specified.
As given in the question,
Given inequality is equal to :
1 ≤ x ≤ 4
Domain of the given inequality function is given by :
x belongs to all real numbers as it is not specified which set of numbers x belongs.
Consider x as set of real numbers
Given domain is {1,2,3,4}
Error is Felipe needs to specify x must belongs to integers or natural numbers then only domain is {1,2,3,4} for the given inequality else there are infinite many numbers between 1 to 4.
Therefore, for the inequality 1 ≤ x ≤ 4 given by Felipe the domain of the function is stated as {1,2,3,4} which shows the error of x belongs to which set of numbers is not specified.
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1. Which of the following is not a radical expression that is equivalent to √1087A.B.C.D.√2-√54√3-√36√5-√21√6-√18
The radical expression √108 is not equivalent to the expression √5√21 .
Given the radical expression as √108 .
Now the number 108 can be broken down into factors as
108 = 54 × 2 , 36 × 3 , 18×6
therefore we can see that the radical expression is equivalent to
√108 = √54 × √2
√108 = √36 × √3
√108 = √18 × √6
But 21 × 5 = 105 ≠ 108.
Therefore the radical expression √108 is not equivalent to √5√21 .
Expressions in mathematics are statements with variables, numbers, or both, and at least two terms joined by an operator. Mathematical operations include addition, subtraction, multiplication, and division.
In mathematics, there are two types of expressions: numerical expressions, which only contain numbers, and algebraic expressions, which also contain variables.
A symbol with an unknown value is called a variable. A term can be made up of a single constant, a single variable, or a group of variables and constants multiplied or divided. A number that has been further multiplied by a variable serves as the coefficient in an equation.
Disclaimer: The complete question is :
Which of the following is not a radical expression that is equivalent to √108?
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The local seven-digit telephone numbers in city A have 1,8,0, as the first three digits. How many different telephone numbers are possible in city A?
ANSWER
10,000
EXPLANATION
Given:
The first-three digits to be 1, 8, 0 out of seven-digit telephone numbers.
Desired outcome:
Total number of possible different telephone numbers
Determine the possibilities of the 4th, 5th, 6th and the 7th digits
The 4th digit has 10 possibilities (i.e: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
The 5th also has 10 possibilities,
likewise the 6th and the 7th digits.
Now, we have:
10 x 10 x 10 x 10 = 10^4 = 10,000
Hence, the number of possible different telephone numbers is 10,000.
For each line the SLOPE between the 2 points given - simplify each fraction to prove that the lines have a CONSTANT rate of change : 1) Point T : 2) Point R : 3) Point S : 4) Slope of TR : 5) Slope of RS : 6) Slope of TS : 7) Describe the SLOPE of the line : 8) Therefore the CONSTANT RATE OF CHANGE IS ...?
the point T on the line is T(-7,6)
point R = R(-3,0)
point S = S(1,-6)
the slope of TR is
[tex]\begin{gathered} m=\frac{6-0}{-7-(-3)} \\ m=-\frac{6}{4} \\ m=-\frac{3}{2} \end{gathered}[/tex]slope of RS,
m = (0 - (-6))/(-3-1)
= - 6/4
= -3/2
slope of TS
m = (-6-6)/ 1-(-7)
= -12/ 8
= -3/2
the slope of the line or the constant rate of change is m = -3/2
the u.s senate has 100 members. after a certain election, there were 6 more democrats than republicans, with no other parties represented. how many memebrs of each party were there in the senate?
As per the unitary method, the number of democrats are 53 and the number of republicans are 47.
Unitary method:
Unitary method is the process of finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
Given,
The U.S senate has 100 members. after a certain election, there were 6 more democrats than republicans, with no other parties represented.
Here we need to find the number of democrats and the number of republicans.
Let us consider the number of Republican's be "x"
And then the number of Democrats is "x+6".
We know that the total number of members in the senate is 100.
So, it can be written as
x + (x + 6) = 100
2x + 6 = 100
2x = 100 - 6
2x = 94
x = 94/2
Then the value of x is,
x = 47
Therefore, there are 47 republicans in the senate then the number of democrats is
x + 6 => 47 + 6 => 53
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Which relationship can be represented by the equation y = 1/5x A) One wager jug hold 5 quarts. Let x represent the number of water jugs and y represent the number of quartsB) Bananas are on sale for $0.20 per banana. Let x represent the number of bananas and y represent the total cost, in dollars.C) David runs ar a constant rate. He runs 15 mines in 3 hours. Let x represent the number of hours and y present the number of miles.D) For every 10 gallons of water, Jasmine adds 2 cups of soap. Let x represent the number of hours and y represent the number of miles.E) The library charges a few at a rate of $5 for every 10 days a book is late. Let x represent the number of days late and y represent the fee, in dollars.
Answer:
Explanation:
Option A
[tex]\begin{gathered} \text{One wager jug hold 5 quarts} \\ y=kx \\ 5=1k \\ y=5x \end{gathered}[/tex]Option B
[tex]\begin{gathered} 1\text{ banana costs \$}$0.20$ \\ y=0.20x \\ y=\frac{1}{5}x \end{gathered}[/tex]The correct choice is B.
Find the coordinates of the missing vertex of rectangle ABCD with A(-3, 3),B(5, 3), and D(-3, -1).O (5, -1)© (-11, 7)O (5,3)O (1, -1)
Given:
The given vertex of a rectangle ABCD are A=(-3,3), B=(5,3) and D=(-3,-1)
To find: Missing vertex, that means vertex C
The graph is as follows:
From the above graph, the coordinates of point C are (5,-1).
Hence, the required answer is (5,-1).
A If mzABD 61, and mzDBC = 59, then mABC = [ ?P
The population P of a city is given by P = 115600e^0.024t, where t is the time in years. According to this model, after how many years will the population be 130,000?4.29 years4.89 years5.19 years4.49 years
Given:
The population P of a city is given by,
[tex]P=115600e^{0.024t,}[/tex]To find:
The time taken for the population to reach 130,000.
Explanation:
Substituting P = 130,000 in the given function, we get
[tex]\begin{gathered} 130000=115600 \\ e^{0.024t}=\frac{130000}{115600} \\ e^{0.024t}=1.1245 \\ 0.024t=\ln1.1245 \\ 0.024t=0.1174 \\ t=4.891 \\ t\approx4.89years \end{gathered}[/tex]Therefore, the number of years required for the population to reach 130,000 is 4.89years.
Final answer:
The number of years required is 4.89years.
Find the measure of each angle in the proplem RE contains point P
In the given figure, line LP is lie on the line RE, thus from the linear property
Angle LPR + Angle LPE = 180°
It is given that Angle LPR = 3z, angle LPE = 2z
Angle LPR + Angle LPE = 180°
3z° + 2z° = 180°
5z° = 180°
z = 180/5
z=36
Substitute the value of z = 36 in the angle LPR;
Angle LPR = 3z
= 3(36)
=108°
Angle LPE = 2z
= 2(36)
= 72°
Answer : B) 108 and 72
Write the slope-intercept form of the equation of each line.3) 10 = -2y-x
Recall that the slope-intercept form of the line equation is of the form y=mx+b, where m is the slope and b is the y-intercept.
To transform the equation 10=-2y-x into the slope-intercept form we should apply algebraic operations so we isolate the y on one side of the equation.
Let's add x on both sides, we get
[tex]-2y=10+x[/tex]Now, lets divide by -2 on both sides, we get
[tex]y=\frac{10}{-2}+\frac{x}{-2}=-\frac{1}{2}\cdot x-5[/tex]we see that this now has the slope-intercept form, where the slope is m=(-1/2) and b=-5
10^4 divided by 10^6 in expanded form, then find out what the answer is with a single power.
The given expression is:
[tex]\frac{10^4}{10^6}[/tex]Exapanding the above expression,
[tex]\frac{10\times10\times10\times10}{10\times10\times10\times10\times10\times10}[/tex](The number of 10s are the same as the power of 10).
Now, cancel out the common terms in the numerator and the denominator of the above expression.
[tex]\frac{1}{10\times10}[/tex]The denominator of the above expression can now be expressed as the power of 10 as,
[tex]\frac{1}{10^2}[/tex]According to the law of exponents,
[tex]\frac{1}{x_{^{^m}}}=x^{-m}[/tex]Hence, we can write
[tex]\frac{1}{10^2}=10^{-2}[/tex]Therefore, 10^4 divided by 10^6 can be expressed as a term with a single power as,
[tex]10^{-2}[/tex]To find the missing length below, would you use Law of Sines or Law of Cosines?Find the missing length. There are 2 answers, Law of _____ and missing side length.
The number of students showing up for a high school football team is 10% smaller than the previous year. A few minutes before tryouts begin, another 5 students show up. There are 75 students on the field to try out for football. Which equation represents this situation?
Given data:
The expression for the given statement is,
[tex]0.9x+5=75[/tex]Thus, the option (C) is correct.
Food Express is running a special promotion in which customers can win a free gallon of milk with their food purchase if there is a star on their receipt. So far, 219 of the first 264 customers have not received a star on their receipt. What is the experimental probability of winning a free gallon of milk?options: 3/11....15/88....73/88.....1/78
Solution:
Experimental probability is a probability that is determined on the basis of a series of experiments. A random experiment is done and is repeated many times to determine their likelihood, and each repetition is known as a trial.
[tex]\begin{gathered} P(E)=\frac{n(E)}{n(T)} \\ \\ Where; \\ n(E)=\text{ number of event} \\ \\ n(T)=\text{ total outcome} \end{gathered}[/tex]If 219 of the first 264 customers have not received a star on the receipt, then a customer that would win a free gallon of milk would be among (264 - 219) customers. Thus;
The experimental probability of winning a free gallon of milk is;
[tex]\begin{gathered} =\frac{264-219}{264} \\ \\ =\frac{45}{264} \\ \\ =\frac{15}{88} \end{gathered}[/tex]CORRECT OPTION:
[tex]\frac{15}{88}[/tex]
cellusTranslate the triangle.Then enter the new coordinates.A -1,61B(0,4)A'([?], [])B'([],[])C'([],[])C (-6,1)< 10.2 >IEnter
The coordinates of the images are;
[tex]undefined[/tex]Here, we want to translate the given triangle
The translation is in the units of (10,2)
What this mean is that we are going to make a translation of 10 in the x-axis direction and 2 units in the y-axis direction
Hence, we are having a unit of 10 rightwards (positive x-axia) and 2 unit of 2 units upwards (vertically on the y-axis)
So what we do now in respective cases is add 10 to the x-axis values of each point and add 2 to the y-axis values of each point
Thus, we have;
[tex]\begin{gathered} A(-1,6)\text{ to A'(-1+10, 6+2)= A'(9,8)} \\ B(0,4)\text{ to B'(0+10, 4+2) = B'(10,6)} \\ C(-6,1)\text{ to C'(-6+10,1+2) = C'(4,3)} \end{gathered}[/tex]Simplity the expression:4b+9b
Since both variables are equal (b) we can add them:
[tex]4b+9b[/tex][tex]13b[/tex]solve the problem and show your work below. I have a rectangular garden. i usually grow cucumbers in2/3 of my garden but i want to take 3/4 of the cucumber section to grow radishes. After i make the change, how much of my whole garden will be radishes?
Given data;
The area in which cucumbers usually grown are 2/3 x.
Here, x is the total area of the garden.
The area of the cucumber taken to grow raddish are,
[tex]\begin{gathered} R=\frac{3}{4}\times\frac{2}{3}x \\ =\frac{1}{2}x \end{gathered}[/tex]Thus, the area of the raddish is 50% of the total area of the garden sfter the changes.
Convert 77.6% to an
equivalent decimal.
Answer:
Step-by-step explanation:
0.776 in decimal form.
Use the table of values for f and g below to find the indicated compositions. (f \circ g)(8) =Answerg(f(3))=Answerf(f(1))=Answer (g \circ g)(6) =Answer
In order to find the value of a composition of functions at x = a, (f º g)(a), we first find the value of g(a), then find the value of f(x) at x = g(a).
(f º g)(a) = f(g(a))
In this problem, the values of f(x) and g(x) are shown in the table, for integer values of x from 0 to 9.
So, we have:
1. (f º g)(8) = f(g(8))
From the table, we see that
g(8) = 4 (value of g(x) in the line corresponding to x = 8)
Then:
f(g(8)) = f(4) = 4 (value of f(x) in the line corresponding to x = 4)
Thus:
[tex]\mleft(f\circ g\mright)\mleft(8\mright)=4[/tex]2. g(f(3))
f(3) = 8
g(8) = 4
Thus:
[tex]g(f(3))=4[/tex]3. f(f(1))
f(1) = 6
f(6) = 2
Thus:
[tex]f(f(1))=2[/tex]4. (g º g)(6) = g(g(6))
g(6) = 7
g(7) = 3
Thus:
[tex]\mleft(g\circ g\mright)\mleft(6\mright)=3[/tex]
For each row of the table, choose the equivalent expression
Ok, so:
Let's make all operations and then choose the equivalent expression for each one.
Let's start in order:
a. 4/12 + 4/12 = 8/12
b. 1/12 + (3/12 + 3/12) = 7/12
c. 4/12 + 5/12 = 9/12
d. 2/12 + 2/12 + 2/12 = 6/12.
Notice that the last operations are the columns of the table.
So, let's do it the same with the upper rows:
e. 5/12 + 4/12 = 9/12
f. (1/12 + 3/12) + 3/12 = 7/12
g. 1/12 + 2/12 + 3/12 = 6/12
h. 15/12 - 7/12 = 8/12.
Now, let me draw the table to make this problem more understandable.
This is the order you have to put the answer:
Aiden ipens a savings account with a deposit of 4500. The account pays 3% simple interest.3. If Aiden does not make any more deposits or withdrawals, how much will he have in the account at the end of two years?A 4527B 4635C 4680D 4774E 4905
Answer: $4, 770
Aiden deposit $4500 into her account with an interest rate of 3%
Time = 2 years
Using the Simple Interest
[tex]\begin{gathered} I\text{ = }\frac{P\text{ x R x T}}{100} \\ P\text{ = \$4500} \\ R\text{ = 3\%} \\ T\text{ = 2} \\ I\text{ = }\frac{4500\text{ x 3 x 2}}{100} \\ I\text{ = }\frac{4500\text{ x 6}}{100} \\ I\text{ = }\frac{27000}{100} \\ I\text{ = \$270} \\ \text{The total amount in her account is } \\ \text{Balance = Principal + Interest} \\ \text{Balance = \$4500 + \$270} \\ \text{Balance = \$4, 770} \end{gathered}[/tex]Write the coordinates of the vertices after a rotation of 90 degrees counter clock wise around the origin. Give me the coordinates and that’s it no explanation
An empty shipping box weighs 250 grams. The box is then filled with t-shirts. Each tshirts weighs 132.5 grams. The equation W = 250 + 132.5T represents the relationship between the quantities in this solution where W is the weight in grams of the filled box and T the number of shirts in the box. Consider this equation 2900 = 250 + 132.5T. What does the solution to this equation tell us?
Given the next equation
2900 = 250 + 132.5T
its solution is:
2900 - 250 = 132.5T
2650 = 132.5T
2650/132.5 = T
T = 20
Given that W is weigth and T is t-shirts, the solution tell us that a box with 20 t-shirts weights 2900 grams
Which is a solution toy ⩽ -2x + 1 (-3,8)(2,-2)(0,5)(-1, 3)
To find the right solution, we just have to evaluate the expression with each given point
(-3,8)[tex]\begin{gathered} 8\leq-2\cdot(-3)+1 \\ 8\leq6+1 \\ 8\leq7 \end{gathered}[/tex](2,-2)[tex]\begin{gathered} -2\leq-2\cdot2+1 \\ -2\leq-4+1 \\ -2\leq-3 \end{gathered}[/tex](0,5)[tex]\begin{gathered} 5\leq-2\cdot0+1 \\ 5\leq1 \end{gathered}[/tex](-1,3)[tex]\begin{gathered} 3\leq-2\cdot(-1)+1 \\ 3\leq2+1 \\ 3\leq3 \end{gathered}[/tex]As you can observe, the last choice satisfies the inequality.
Hence, the answer is (-1,3).Perform the indicated operation -27÷-9
-27/9 = -3
answer is -3
Which set of polar coordinates names the same point as (-5.5) ? ZT O O A. (5, O B. (5:59) O 5 57 4 377 O c. -5 O D. 7T 5. )
Recall that the following points represent the same point as the point (x,θ)
[tex]\begin{gathered} (-x,\theta+\pi), \\ (-x,\theta-\pi), \\ (x,\theta+2n\pi)\text{.} \end{gathered}[/tex]Now, notice that:
[tex]\frac{5\pi}{4}=\frac{4\pi}{4}+\frac{\pi}{4}=\pi+\frac{\pi}{4}\text{.}[/tex]Therefore, the point:
[tex](5,\frac{5\pi}{4})[/tex]represent the same point as the point
[tex](-5,\frac{\pi}{4})\text{.}[/tex]Answer: Option B.
